The objective was to estimate (co)variance functions using random regression models (RRM) with Legendre polynomials, B-spline function and multi-trait models aimed at evaluating genetic parameters of growth traits in meat-type quail. A database containing the complete pedigree information of 7000 meat-type quail was utilized. The models included the fixed effects of contemporary group and generation. Direct additive genetic and permanent environmental effects, considered as random, were modeled using B-spline functions considering quadratic and cubic polynomials for each individual segment, and Legendre polynomials for age. Residual variances were grouped in four age classes. Direct additive genetic and permanent environmental effects were modeled using 2 to 4 segments and were modeled by Legendre polynomial with orders of fit ranging from 2 to 4. The model with quadratic B-spline adjustment, using four segments for direct additive genetic and permanent environmental effects, was the most appropriate and parsimonious to describe the covariance structure of the data. The RRM using Legendre polynomials presented an underestimation of the residual variance. Lesser heritability estimates were observed for multi-trait models in comparison with RRM for the evaluated ages. In general, the genetic correlations between measures of BW from hatching to 35 days of age decreased as the range between the evaluated ages increased. Genetic trend for BW was positive and significant along the selection generations. The genetic response to selection for BW in the evaluated ages presented greater values for RRM compared with multi-trait models. In summary, RRM using B-spline functions with four residual variance classes and segments were the best fit for genetic evaluation of growth traits in meat-type quail. In conclusion, RRM should be considered in genetic evaluation of breeding programs.

The test-day yields of milk, fat and protein were analysed from 1433 first lactations of buffaloes of the Murrah breed, daughters of 113 sires from 12 herds in the state of São Paulo, Brazil, born between 1985 and 2007. For the test-day yields, 10 monthly classes of lactation days were considered. The contemporary groups were defined as the herd-year-month of the test day. Random additive genetic, permanent environmental and residual effects were included in the model. The fixed effects considered were the contemporary group, number of milkings (1 or 2 milkings), linear and quadratic effects of the covariable cow age at calving and the mean lactation curve of the population (modelled by third-order Legendre orthogonal polynomials). The random additive genetic and permanent environmental effects were estimated by means of regression on third- to sixth-order Legendre orthogonal polynomials. The residual variances were modelled with a homogenous structure and various heterogeneous classes. According to the likelihood-ratio test, the best model for milk and fat production was that with four residual variance classes, while a third-order Legendre polynomial was best for the additive genetic effect for milk and fat yield, a fourth-order polynomial was best for the permanent environmental effect for milk production and a fifth-order polynomial was best for fat production. For protein yield, the best model was that with three residual variance classes and third- and fourth-order Legendre polynomials were best for the additive genetic and permanent environmental effects, respectively. The heritability estimates for the characteristics analysed were moderate, varying from 0·16±0·05 to 0·29±0·05 for milk yield, 0·20±0·05 to 0·30±0·08 for fat yield and 0·18±0·06 to 0·27±0·08 for protein yield. The estimates of the genetic correlations between the tests varied from 0·18±0·120 to 0·99±0·002; from 0·44±0·080 to 0·99±0·004; and from 0·41±0·080 to 0·99±0·004, for milk, fat and protein production, respectively, indicating that whatever the selection criterion used, indirect genetic gains can be expected throughout the lactation curve.

The objective of this study was to estimate (co)variance components using random regression on B-spline functions to weight records obtained from birth to adulthood. A total of 82 064 weight records of 8145 females obtained from the data bank of the Nellore Breeding Program (PMGRN/Nellore Brazil) which started in 1987, were used. The models included direct additive and maternal genetic effects and animal and maternal permanent environmental effects as random. Contemporary group and dam age at calving (linear and quadratic effect) were included as fixed effects, and orthogonal Legendre polynomials of age (cubic regression) were considered as random covariate. The random effects were modeled using B-spline functions considering linear, quadratic and cubic polynomials for each individual segment. Residual variances were grouped in five age classes. Direct additive genetic and animal permanent environmental effects were modeled using up to seven knots (six segments). A single segment with two knots at the end points of the curve was used for the estimation of maternal genetic and maternal permanent environmental effects. A total of 15 models were studied, with the number of parameters ranging from 17 to 81. The models that used B-splines were compared with multi-trait analyses with nine weight traits and to a random regression model that used orthogonal Legendre polynomials. A model fitting quadratic B-splines, with four knots or three segments for direct additive genetic effect and animal permanent environmental effect and two knots for maternal additive genetic effect and maternal permanent environmental effect, was the most appropriate and parsimonious model to describe the covariance structure of the data. Selection for higher weight, such as at young ages, should be performed taking into account an increase in mature cow weight. Particularly, this is important in most of Nellore beef cattle production systems, where the cow herd is maintained on range conditions. There is limited modification of the growth curve of Nellore cattle with respect to the aim of selecting them for rapid growth at young ages while maintaining constant adult weight.

Let R(t) be the covariance function of the stationary virtual waiting-time process of a stable M/G/1 queue. It is proven that if R(t) exists, i.e., if the service-times have a finite third moment, then R(t) is positive and convex on [0, ∞), with an absolutely continuous derivative R’ and a bounded, non-negative second derivative R″. Also, and R″ cannot be chosen monotone. Contrary to a finding by Beneš [1] it is proven that if and only if the service-times have a finite fourth moment.

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}.

For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t).

These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

We consider an important class of integer valued stochastic processes characterised by a certain form of time dependent probability generating function (P.G.F.). Certain results are obtained concerning the correlation structure of such processes, and it is shown that these are considerably simplified in the case of a stationary distribution having been attained. The results are illustrated in the latter case using some commonly occurring processes belonging to the class under consideration.

Several authors have considered the covariance structure of continuous parameter Markov chains. Most of this work has dealt with particular process ses, notably Morse (1955) who analysed the simple M/M/1 queue and Bene-(1961) who considered a telephone trunking model. Furthermore, the results obtained apply only when the process has attained its limiting (stationary) distribution. A recent paper by Reynolds (1968) gave some general results for finite chains, still assuming stationarity. This note generalises the results obtained therein, and considers the covariance structure during the transient period prior to attaining the stationary distribution where this exists. In the case where no such distribution exists, the results are valid throughout the whole lifetime of the process.